3.1 Recognizing Discrete Probability Models
Key Terms
- Probability model（機率模型）
A mathematical description of the possible values of a random variable and their probabilities.
- Parameter（參數）
A numerical value that controls the form of a probability distribution.
- Bernoulli trial（伯努利試驗）
A trial with two possible categories, commonly labeled success and failure.
- Success（成功）
The outcome being counted in a Bernoulli-type experiment; the word is only a label.
- Failure（失敗）
The other outcome in a Bernoulli-type experiment.
- Rate（發生率）
The average number of events occurring per unit of time, distance, area, or volume.
3.1.2 Learning Outcomes
After completing this chapter, students should be able to:
explain why the same named distribution can model many different experiments;
identify the assumptions of common discrete probability distributions;
identify the random variable and its range before using a formula;
calculate simple probabilities from discrete uniform, binomial, geometric, negative binomial, hypergeometric, and Poisson distributions;
calculate or interpret the mean and variance of these distributions;
distinguish fixed-number-of-trials problems from waiting-until-success problems;
distinguish sampling with replacement from sampling without replacement;
distinguish counting successes in trials from counting events in time or space;
choose an appropriate discrete probability model from the structure of a problem.
3.1.3 Why Named Distributions Are Useful
In Chapter 2, a discrete probability distribution was described by a PMF:
For a new experiment, we could construct the entire distribution directly from the sample space.
However, many different experiments have the same probability structure.
For example:
number of heads in 10 coin tosses;
number of correct answers when guessing on 10 questions;
number of defective items among 10 independently produced items.
The contexts are different, but the mathematical structure can be the same.
When several experiments share the same structure, one named probability distribution can be used for all of them.
The main idea is
recognize the experiment structure
-> choose a probability distribution
-> identify its parameters
-> calculate the required probability.
3.1.4 Distribution Formulas Are Not Chosen by Keywords Alone
A probability distribution should be selected from the assumptions of the experiment.
For example, the phrase number of defective items does not automatically
mean binomial.
If 5 items are produced independently from a process with a constant defective probability, a binomial model may be appropriate.
If 5 items are sampled without replacement from a small lot containing a known number of defective items, a hypergeometric model is more appropriate.
Therefore, before using a formula, ask:
What does \(X\) count?
Is the number of trials fixed?
Are trials independent?
Is the probability of success constant?
Is sampling with or without replacement?
Are we counting events in a time or spatial interval?
Are all possible values equally likely?
3.1.5 A Preview of the Main Models
Distribution |
Main Random Variable |
Key Structure |
|---|---|---|
Discrete uniform |
One value from a finite set |
All possible values are equally likely |
Binomial |
Number of successes |
Fixed number of independent Bernoulli trials |
Geometric |
Trial number of first success |
Repeat Bernoulli trials until first success |
Negative binomial |
Trial number of the \(r\)-th success |
Repeat Bernoulli trials until \(r\) successes |
Hypergeometric |
Number of successes in a sample |
Finite population sampled without replacement |
Poisson |
Number of events in an interval |
Events occur randomly at a specified average rate |
3.1.6 Common Mistake: Formula First
A common mistake is to see a probability question and immediately search for a formula.
The safer approach is:
define \(X\)
-> identify the experiment structure
-> check the assumptions
-> identify the parameters
-> use the formula.
A correct formula applied to the wrong experiment produces a wrong model.